The space-time fractional diffusion equation with Caputo derivatives

  • F. Huang
  • F. Liu


We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

AMS Mathematics Subject Classification

26A33 49K20 44A10 

Key words and phrases

Fractional diffusion equation time-space Caputo derivative Fourier transform Laplace transform Mittag-Leffler function Green function stable probability distributions 


  1. 1.
    O. P. Agrawal,Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics29 (2002), 145–155.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    V. V. Anh and N. N. Leonenko,Non-Gaussian scenarios, for the heat equation with singular initial conditions, Stochastic Processes and their Applications84 (1) (1999), 91–114.CrossRefMathSciNetGoogle Scholar
  3. 3.
    V. V. Anh and N. N. Leonenko, Scaling laws for fractional diffusion-wave equations with singular data, Statistics and Probability Letters Volume48 (3), (2000), 239–252.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    V. V. Anh and N. N. Leonenko,Spectral analysis of fractional kinetic equations with random data, J. Stat. Physics,104, N5/6 (2001), 1349–1387.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    V. V. Anh and N. N. Leonenko,Renormalization and homogenization of fractional diffusion equations with random data, Probab. Theory Rel. Fields124 (2002), 381–408.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    V. V. Anh and N. N. Leonenko,Harmmonic analysis of fractional diffusion-wave equations, Applied Math. Comput.48 (3) (2003), 239–252.MathSciNetGoogle Scholar
  7. 7.
    M. Basu and D. P. Acharya,On quadratic fractional generalized solid bi-criterion, J. Appl. Math. & Computing (old: KJCAM)10 (2002), 131–144.MATHMathSciNetGoogle Scholar
  8. 8.
    M. Caputo,Linear model of dissipation whose Q is almost frequency indepent-II, Geophys. J. R. Astr. Soc.13 (1967), 529–539.Google Scholar
  9. 9.
    M. Caputo,The Green function of the diffusion of fluids in porous media with memory, Rend, Fis. Acc. Lincei (Ser. 9)7 (1996), 243–250.MATHCrossRefGoogle Scholar
  10. 10.
    M. M. Djrbashian,Integral transforms and representations of functions in the complex plane, Nauka, 1966 (in Russian).Google Scholar
  11. 11.
    A. M. A. El-Sayed and M. A. E. Aly,Continuation theorem of fractionalorder evolutionary integral equations, J. Appl. Math. & Computing (old: KJCAM)9 (2) (2002), 525–534.MATHMathSciNetGoogle Scholar
  12. 12.
    A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,Higer transcendental functions, New York, McGraw-Hill Vol. 3, 1953-1954.Google Scholar
  13. 13.
    W. Feller,On a generalization of Marcel Riesz's potentials and the semigroups generated by them, Meddelanden lunds Universitets Matematiska Seminarium (Comm. Sém. M..a.thém. Université de Lund), Tome suppl. dédié à M. Riesz, Lund, pp. 73–81, 1952.Google Scholar
  14. 14.
    Y. Fujita,Integrodifferential equation which interpolates the heat equation and the wave equation, Osaka, J. Math.27 (1990), 309–321.MATHMathSciNetGoogle Scholar
  15. 15.
    A. A. Kilbas, T. Pierantozzi and J. Trujillo,On the solution of fractional evolution equations, J. Phys. A: Math. Gen.37 (2004), 3271–3283.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    R. Gorenflo, Yu. Luchko and F. Mainardi,Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math.118 (2001), 175–191.CrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Gorenflo and F. Mainardi,Fractional calculus: integral and differential equations of fractional order, in A. Carpinteri and Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, Wien and New York, Springer Verlag., pp. 223–276, 1997.Google Scholar
  18. 18.
    R. Gorenflo and F. Mainardi,Random walk models for space-fractional diffusion processes, Fractional Calculus and Applied Analysis1, (1998), 167–191.MATHMathSciNetGoogle Scholar
  19. 19.
    R. Gorenflo and F. Mainardi,Approximation of Lévy-Feller diffusion by random walk, Journal for Analysis and its Applications (ZAA)18 (1999), 231–246.MATHMathSciNetGoogle Scholar
  20. 20.
    R. Gorenflo and F. Mainardi, D. Moretti, G. Pagnini,Time-fractional diffusion: a discrete random walk approach, Nonlinear Dynamics29 (2002), 129–143.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    R. Gorenflo and F. Mainardi, D. Moretti and G. Pagnini,Disctete random walk models for space-time fractional diffusion, Chemical Physics284 (2002), 521–541.CrossRefGoogle Scholar
  22. 22.
    R. Hilfer,Exact solutions for a class of fractal time random walk, Fractals3 (1995), 211–216.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    F. Huang and F. Liu,The time fractional diffusion equation and advection-dispersion equation, the Australian and New Zealand Industrial and Applied Mathematic Journal (ANZIAM), (2004), in press.Google Scholar
  24. 24.
    F. Liu, V. V. Anh, I. Turner and P. Zhuang,Time Fractional Asvection-dispersion Equation, J. Appl. Math. & Computing13 (2003), 233–245.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    F. Liu, V. V. Anh and I. Turner,Numerical, solution of the space fractional Fokker-Plank Equation., J. Comp. Appl. Math166 (2004), 209–219.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    F. Mainardi,Fraction calculus: some basic problems in continuum and statistical mechanics (A. Carpinteri, F. Mainardi, Eds.),Fractal and Fractional Calin Continuum Mechanics, Springer, Wien, pp. 291–348, 1997.Google Scholar
  27. 27.
    F. Mainardi, Yu. Luchko and G. Pagnini,The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis4 (2001), 153–1925.MATHMathSciNetGoogle Scholar
  28. 28.
    M. M. Meerschaert and X. Tadjeran,Finite difference approximations for fractional advection-dispersion flow, equations, J. Appl. Math. and Computing, (2004) in press.Google Scholar
  29. 29.
    P. J. Olver,Applications of Lie Groups to Differential Equations, Springer, New York, 1986.MATHGoogle Scholar
  30. 30.
    I. Podlubny,Fractional differential equations, Academic press, San Diego, 1999.MATHGoogle Scholar
  31. 31.
    A. Saichev and G. Zaslavsky,Fractional kinetic equations: solutions and applications, Chaos7 (1997), 753–764.MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    W. R. Schneider and W. Wyss,Fractional diffusion and wave equations, J. Math. Phys.30 (1) (1989), 134–144.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    R. Schumer, D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft,Eulerian derivation of the fractional advection-dispersion equation, J. Contaminant Hydrology48 (2001), 69–88.CrossRefGoogle Scholar
  34. 34.
    W. Wyss,The fractional diffusion equation., J. Math. Phys.27 (11) (1986), 2782–2785.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.School of Mathematical SciencesQueensland University of TechnologyAustralia
  3. 3.School of Mathematical SciencesSouth China University of TechnologyGuangzhouChina

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